The efficiency of one fluid displacing another in a permeable medium depends on the pore-scale dynamics at the main wetting front. Experiments have shown that the frontal dynamics can result in two different flow regimes: a sharp and a diffuse front. In the sharp front regime, the displacing fluid occupies nearly all the pores and throats behind the main wetting front and the saturation changes abruptly. In contrast, in the diffuse front regime, pores are filled gradually at the main wetting front, and the saturation change is gradual in space. The different fronts can greatly alter the relative permeability curves, the trapping mechanisms, and the displacement efficiency. Directly measuring the sharpness of the front is difficult. Instead, we correlated the front sharpness to saturation overshoot, which occurs for moderate- to high-flux vertical displacements of low-density fluid by a higher density fluid in one-dimensional, homogenous, permeable media. We hypothesized that the sharpness of wetting front can be explained by competition between two different pore-filling mechanisms (called snap off and piston like), with the competition controlled by the velocity of the front and thus the injected flux. We conducted a series of infiltration experiments to determine the saturation profile as a function of flux for seven different fluids. We found that for each fluid there is a flux (called overshoot flux) below which saturation overshoot ceases and the front is diffuse. We found that the overshoot flux depends inversely on the invading fluid's viscosity and shows little or no dependence on the invading fluid's surface tension, vapor pressure, or miscibility with water.